Unit:1.1 Rational Numbers Days :20 Essential Questions How do you write rational numbers as decimals and as fractions? How do you multiply rational numbers? How do you divide rational numbers? How do you add and subtract fractions with unlike denominators? How do you solve equations that contain rational numbers? How do you solve equations that contain multiple operations? Content to be Learned Skills 1.1 Rational Numbers 1.2 Multiplying Rational Numbers 1.3 Dividing Rational Numbers 1.4 Adding and Subtracting with Unlike Denominators 1.5 Solving Equations with Rational Numbers 1.6 Solving Two-Step Equations Assessments 1.1/1.2/1.3 Quiz (9 days) 1.4/1.5 Quiz (7 days) 1.6 Unit Test (4 days) Sample Instructional Activities Converting Decimal Representations of Rational Numbers to Fraction Representations Converting Repeating Decimals to Fractions Writing rational numbers as decimals and fractions as well as multiplying and dividing rational numbers. The sum and difference of fractions with unlike denominators will be found using a common denominator as well as solving equations with rational numbers. Learn to identify and apply the appropriate property of equality for a given real world situation. Practice writing and solving two-step equations for real world situations. Standards 8.NS.1 8.EE.7b Resources Explorations in Core Math Illustrative Mathematics
Unit: 1.1 Multiplying and Dividing Rational Numbers Lesson 1 of 2 Days 9 Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.NS.1 8.EE.7b Writing rational numbers as decimals and fractions as well as multiplying and dividing rational numbers. Standard 2. Reason abstractly and quantitatively. Standard 7. Look for and make use of structure. How do you write rational numbers as decimals and as fractions? How do you multiply rational numbers? How do you divide rational numbers? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Converting fractions into decimals. Rational numbers Rewriting mixed numbers as improper fractions Rewriting improper fractions as mixed numbers Rational number Irrational number Repeating decimal Terminating decimal Reciprocals Inverse The difference between a terminating and non-terminating decimal. Not multiplying by the reciprocal when dividing. How many places to move the decimal point when multiplying or dividing. Translating from horizontal form to the division house form. (not knowing the difference between the dividend and the divisor) Explorations in CORE MATH Converting Decimal Representations of Rational Numbers to Fraction Representations Converting Repeating Decimals to Fractions
Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Rational Numbers Show QR Reader video and then read the Essential question. Ask students to brainstorm some rational numbers. Direct students to do Example 1. Have the students pay special attention to whether or not their decimal terminates or repeats. Have students do Example 2. (may want to do a quick mini lesson on repeating decimals and their fraction representation) Have students practice some problems and close with the essential question. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you write rational numbers as decimals and as fractions? Student led discussion on the rational numbers they have shared with the class. Students will work on Example 1 in their groups and follow with a student led discussion on the difference between a terminating and repeating decimal. Students will discuss their results and share them with their classmates. Students will work on their practice problems and then discuss the Essential question; how do you write rational numbers as decimals and as fractions? Lesson on Multiplying Rational Numbers Show QR Reader video and then read the Essential question. Remind students that whole numbers/integers can be written as fractions with a denominator of 1. Have students do Example 1. Remind students about the reasonableness of their solutions. Instruct students to do Example 2. (You may want to do a quick mini lesson on changing mixed numbers to improper fractions and improper fractions to mixed numbers). Also remind students about rules when multiplying positive and negative numbers. Have students practice some problems and close with the essential question. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you multiply rational numbers? Students will work on Example 1 in their groups and discuss whether their solutions make sense. Students work on Example 2 as well as discussing how to go from mixed numbers to improper fractions and improper fractions to mixed numbers. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you multiply rational numbers?
Lesson on Dividing Rational Numbers Show QR Reader video and then read the Essential question. Have students work on Example 1. Remind students that they need to be careful with decimal placement. Can also remind them that they can use multiplication to check their decimal division. Same rules apply when multiplying positive and negative numbers. Instruct students to do Example 2. Remind them when dividing fractions they are actually multiplying by the reciprocal or inverse. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you divide rational numbers? Students will work on Example 1 in their groups and discuss their results. Students work on Example 2 as well as discussing how to check their results. Have students practice some problems and close with the essential question. Remind students about moving the decimal point as well as multiplying by the reciprocal when working with fractions. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you divide rational numbers?
Unit: 1.1 Adding and Subtracting Fractions and Solving Equations Lesson Focus Lesson 2 of 2 Days 11 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.EE.7b The sum and difference of fractions with unlike denominators will be found using a common denominator as well as solving equations with rational numbers. Learn to identify and apply the appropriate property of equality for a given real world situation. Practice writing and solving twostep equations for real world situations. Standard 2. Reason abstractly and quantitatively. How do you add and subtract fractions with unlike denominators? How do you solve equations that contain rational numbers? How do you solve equations that contain multiple operations? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Adding and subtracting with like denominators. Solving equations with integers. Explorations in CORE MATH Numerator Denominator Linear equation Variable Term Coefficient Constant Adding denominators when adding/subtracting fractions. Not finding a common denominator when adding or subtracting. Not following proper steps when solving equations. (not using the properties of equality the correct way) When dividing by a fraction not multiplying by the inverse.
Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Adding and Subtracting with Unlike Denominators Show QR Reader video and then read the Essential question. Instruct students to do Example 1. Remind them that in order to add and subtract fractions with unlike denominators that they must find a common denominator. Ask some groups to share how they arrived at their solutions. Instruct students to practice some problems. Let them know you will be reviewing them one problem at a time as groups complete the task. Close with the Essential question and then ask students for the key points. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you add and subtract fractions with unlike denominators? Students will work on Example 1 in their groups and discuss whether their solutions make sense. Some students will present their results to the class explaining their steps. Students will work on their problems in their groups and present their result to the entire class. Students will brainstorm some key points sharing their results with the class. Lesson on Solving Equations with Rational Numbers Show QR Reader video and then read the Essential question. Instruct students to begin Example 1. Emphasize the importance of defining a variable. Remind them that they are looking to isolate the variable to make the equation true, while using the properties of equality to keep the equation balanced. (Explain that they can check their work by plugging in their solution to the original problem) Instruct students to begin Example 2, once again emphasize the importance of defining a variable. Also give the students a friendly reminder on needing a common denominator when adding or subtracting. Have students do Example 3. Remind them that you undo multiplication by using the opposite operation division. Point out that the word descended indicates a negative amount. Have students do Example 4. Remind them that you undo division by using the opposite operation multiplication. Have students practice some problems and close with the essential question. Remind students about the properties of equality when Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you solve equations that contain rational numbers? Students will work on Example 1. They will share their results. Have a student led discussion on the steps they used to solve the equation. Students will work on Example 2 and share their results with the class. Students will work on the example and share their solutions with the class. Students work on Example 4 sharing their solutions. Students will work on the problems while sharing their solutions
solving equations. and then discuss the Essential question; how do you solve equations that contain rational numbers? Lesson on Solving Two-Step Equations Show QR Reader video and then read the Essential question. Remind students that when solving a two-step equation that you typically undo the operations in reverse order. You undo addition and subtraction before you undo multiplication and division. Instruct students to begin Example 1. (Explain that they can check their work by plugging in their solution to the original problem) Instruct students to begin Example 2, once again emphasize the importance of defining a variable. Have students practice some problems and close with the essential question. Remind students about the properties of equality when solving equations. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you solve equations that contain multiple operations? Students will work on Example 1. They will share their results. Have a student led discussion on the steps they used to solve the equation. Students will work on Example 2 and share their results with the class. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you solve equations that contain multiple operations?
Unit:1.2 Graphs and Functions Days : 12 Essential Questions How do you determine whether an ordered pair is a solution to an equation? How do you locate and name points in the coordinate plane? How can you describe a relationship given a graph and sketch a graph given a description? How do you represent a function with a table or graph? How can you use equations, tables, and graphs to represent relationships between two variables? Content to be Learned Skills 2.1 Ordered Pairs 2.2 Graphing on a Coordinate Plane 2.3 Interpreting Graphs 2.4 Functions 2.5 Equations, Tables, and Graphs Assessments 2.1/2.2 Quiz (3 days) 2.3/2.4 Quiz (5 days) 2.1-2.5 Unit Test (4 days) Sample Instructional Activities http://ccssmath.org/?page_id=707 Graphing Stories Determining whether ordered pairs are solutions to equations as well as locating points on a coordinate plane. Describe relationships given a graph while sketching a graph given a description. Represent functions with a table or graph as well as using equations, tables, and graphs to represent relationships between two variables. Standards 8.EE.8 8.SP.1 8.F.5 8.F.1 8.F.4 Resources Explorations in CORE MATH
Unit: 1.2 Graphs and Functions Lesson 1 of 1 Days 12 Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.EE.8 8.SP.1 8.F.5 8.F.1 8.F.4 Determining whether ordered pairs are solutions to equations as well as locating points on a coordinate plane. Describe relationships given a graph while sketching a graph given a description. Represent functions with a table or graph as well as using equations, tables, and graphs to represent relationships between two variables. Standard 2. Reason abstractly and quantitatively. Standard 6. Attend to Precision. Standard 3. Construct viable arguments and critique the reasoning of others. Standard 4. Model with Mathematics. How do you determine whether an ordered pair is a solution to an equation? How do you locate and name points in the coordinate plane? How can you describe a relationship given a graph and sketch a graph given a description? How do you represent a function with a table or graph? How can you use equations, tables, and graphs to represent relationships between two variables? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Solving one variable equations. Solutions Ordered pairs Coordinate plane Function Linear function nonlinear function Rate of change Input, output Ordered pair Origin Quadrant x-axis, x-coordinate y-axis, y-coordinate Reversing the (x, y) coordinates when graphing. Not labeling graphs the proper way. (time is always represented as the independent variable) Some students may confuse the rate of change (change in y/change in x) and do the opposite. Explorations in CORE MATH Graphing Stories Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Ordered Pairs Show QR Reader video and then read the Essential question. Students will watch the video taking notes. The class will discuss
Instruct students to do Example 1. Remind students that when plugging in the ordered pair (3, 15) (x, y); the value 3 represents the x-value and the value 15 represents the y-value. Thus order in an ordered pair is important. Have students practice some problems and close with the Essential question. Lesson on Graphing on a Coordinate Plane: Show QR Reader video and then read the Essential question. Instruct students to do Example 1. Remind them once again that the coordinate plane is formed by two number lines. The x-axis or horizontal number line and the y-axis or the vertical number line. These axes intersect at right angles to divide the plane into four quadrants. The point where the two axes intersect is the origin and labeled by the ordered pair (0, 0). In an ordered pair (x, y) the first number is the x-coordinate. It tells how many units to move right or left. The y-coordinate tells how many units to move up or down. Have students practice some problems and discuss the Essential question. Lesson on Interpreting Graphs Show QR Reader video and then read the Essential question. the video and the students will highlight the Essential question; how do you determine whether an ordered pair is a solution of an equation? Students do Example 1 showing their solutions to the class. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you determine whether an ordered pair is a solution of an equation? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you locate and name points in the coordinate plane? Students will work on Example 1 and discuss their solutions in a student led discussion. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you locate and name points in the coordinate plane? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you describe a relationship given a graph and sketch a graph given a description? Instruct students to do Explore 1. Point out that time is always graphed on the x-axis. Instruct students to do Explore 2 paying special attention to the three situations in the three graphs. Instruct students to do Explore 3. Students will do Explore 1 as a group and provide their solutions to the class when done. Students will do Explore 2 and share their solutions to the class. Students will do Explore 3 and share their solutions to the class. Students will work on the problems while sharing their solutions and then discuss the Essential question; how can you describe a
Have students practice some problems and close with the Essential question. relationship given a graph and sketch a graph given a description? Lesson on Functions Show QR Reader video and then read the Essential question. Instruct students to do Explore 1. Instruct students to do Explore 2 reminding them that in a table the left column or input represents x, while the right column or output represents y. Instruct students to do Explore 3. Have students practice some problems and close with the Essential question. Lesson on Equations, Tables, and Graphs Show QR Reader video and then read the Essential question. Have students do Example 1. Remind the students that they have used equations to make tables and graphs and will now be moving in the opposite direction to write equations from tables. Have students practice some problems and close with the Essential question. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you represent a function with a table or graph? Students will do Explore 1 as a group and provide their solutions to the class when done. Students will do Explore 2 and share their solutions to the class. Students will do Explore 3 and share their solutions to the class. Students will work on the problems while sharing their solutions and then discuss the Essential question; how do you represent a function with a table or graph? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you use equations, tables, and graphs to represent relationships between two variables? Students will do Example 1 sharing their solutions with their classmates. Students will work on the problems while sharing their solutions and then discuss the Essential question; how can you use equations, tables, and graphs to represent relationships between two variables?
Unit:2-1 Multi-Step Equations Days :15 Essential Questions How do you simplify algebraic expressions? How do you solve equations by collecting like terms and multiplying expressions? How can you give examples of equations with a given number of solutions? How can you solve a system of equations algebraically? Content to be Learned Skills 7.1 Simplify Algebraic Expressions 7.2 Solving Multi-Step Equations 7.3 Solving Equations with Variables on Both Sides 7.4 Systems of Equations Assessments 7.1 Quiz (4 days) 7.2/7.3 Quiz (6 days) 7.4 Quiz/Unit Test (5 days) Sample Instructional Activities 8.EE Solving Equations 8.EE Cell Phone Plans Solve one variable equations with variables on both sides of the equal sign and then determine if the equation has one solution, infinitely many solutions, or no solutions. Understand that the solution to the equation is the value(s) of the variable, which makes a true equality when substituted back into the equation. Using the Distributive Property and combining like terms. Solve systems of two linear equations in two variables algebraically. Solve real-world and mathematical problems leading to two linear equations in two variables. Standards 8.EE.7 8.EE.7a; 8.EE.7b 8.EE.8b; 8.EE.8c Resources Explorations in CORE MATH Illustrative Mathematics
Unit: 2.1 Simplifying Algebraic Expressions and Solving Multi- Step Equations with Variables on Both Sides Lesson Focus Lesson 1 of 2 Days 10 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.EE.7 8.EE.7a 8.EE.7b Write, simplify, and evaluate algebraic expressions. Solve linear equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve equations in one variable with one solution, infinitely many solutions, and no solutions. Standard 1. Make sense of problems and persevere in solving them. Standard 2. Reason abstractly and quantitatively. Standard 4. Model with Mathematics. Standard 7. Look for and make use of structure. How do you simplify algebraic expressions? How do you solve equations by collecting like terms and multiplying expressions? How can you give examples of equations with a given number of solutions? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Algebraic expressions Identifying like terms Solving equations in one variable Solving multi-step equations Like terms Complement Supplement Students turn equations into expressions when working with variables on both sides thus rendering the opposite of the desired solution. Not distributing to all terms inside of the grouping symbols. Explorations in CORE MATH Solving Equations
Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Simplifying Algebraic Expressions Activate prior knowledge and ask students to give some examples of algebraic expressions. Could use QR Reader and show video to class. Ask for a volunteer to read the Essential question. Instruct students to do Example 1 paying special attention to the reflect question and the significance in the context of the problem. Students will give examples of algebraic expressions. After reading the essential question; how do you simplify algebraic expressions, have a student-led discussion on how they have simplified them in the past. Students will do Example 1in their groups and then discuss the significance of what the variable represents in simplified form. Instruct students to do Example 2. Ask students the benefit or advantages of the simplified expression over the original. Have students practice some problems and close with the essential question. Lesson on Solving Multi-Step equations Show QR Reader video and then read the Essential question. Remind students about the importance of combining like terms or simplifying before solving. Instruct students to do Example 1 and Try These examples. Ask students why or if their solution makes sense to Ex. 1. Instruct students to do Explore 2 paying special attention to the keywords used for each expression. Have students practice some problems and close with the essential question. Remind students to define variables and circle keywords when changing verbal phrases to algebraic expressions. Solving Equations with Variables on Both Sides Show QR Reader video and then read the Essential question. Have students do Explore 1. Remind students to use properties of equality with once again the goal being to isolate the variable. Explain to the students that in the past when solving equations they found a single solution and now they will find equations with one Students will do Example 2 in their groups and then discuss the differences of writing a two variable expression to a one variable expression. Students will work on their practice problems and then discuss as a group the Essential question; how do you simplify algebraic expressions? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you solve equations by collecting like terms and multiplying expressions? Students will work on Example 1in their groups and discuss their results with the class. Students will work on example 2 and discuss their solutions with the class. Students will work on their practice problems and then discuss the Essential question; how do you solve equations by collecting like terms and multiplying expressions? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you give examples of equations with a given number of solutions? Students will do Explore 1 in their groups and discuss as they present the 3 different possibilities that can occur when solving
solution, an infinite number of solutions, and no solutions. Have students highlight the KEY Concept and discuss as a class what the chart shows. Have students do Explore 2. Have students do Explore 3. Remind students to define variables and circle keywords before writing their equation. equations. Students will highlight the KEY Concept and discuss the chart. Students will discuss their results for Explore 2. Students will discuss their results for Explore 3. Have students practice some problems and then close with the Essential question asking students for the key points from the lesson. Students will work on their practice problems and then discuss the Essential question; how can you give examples of equations with a given number of solutions?
Unit: 2.1 Systems of Equations Lesson 2 of 2 Days 5 Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.EE.8b 8.EE.8c Solve systems of two linear equations algebraically. Solve real-world mathematical problems in two variables. Standard 5. Use appropriate tools strategically. How can you solve a system of equations algebraically? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Solving systems of equations graphically Slope-intercept form Systems of equations Explorations in CORE MATH Cell Phone Plans Substitution method may be confusing to some students. Not using properties of equality the correct way. Suggested Learning Practices 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Systems of Equations Show QR Reader video and then read the Essential question. Explain to the students that it may be difficult to solve equations with very large values or when the solution has non-integer coordinates. In cases like this solving a system algebraically makes more sense. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you solve a system of equations algebraically? The class will then have a student-led discussion on what may happen when working with very large numbers or when a solution has noninteger solutions. Have students do Example 1 as well as the reflect question when the example is completed. Have students do Example 2 by graphing and then algebraically. Ask the students which way they think is the better method. Omit Example 3. Have students practice some problems and then close with the Essential question asking students for the key points from the lesson. Students will work on the example in their groups and discuss how they can check their answer. Students will work on Example 2. They will then have a class-led discussion on which way they prefer. Students will work on their practice problems and then discuss the Essential question; how can you solve a system of equations algebraically?
Unit:2.2 Graphing Lines Days :22 Essential Questions How do you find rate of change? How can you show that the slope of a line is the same between any two points on the line? How can you use slopes and intercepts to write and graph linear equations? How can you develop the point-slope form of an equation of a line? How can you identify a direct variation? How can you solve a system of equations by graphing? Content to be Learned Skills 8.1 Graphing Linear Equations 8.2 Slope of a Line 8.3 Using Slopes and Intercepts 8.4 Point-Slope Form 8.5 Direct Variation 8.6 Solving Systems of Linear Equations by Graphing Assessments 8.1/8.2 Quiz (7 days) 8.3 Quiz (4 days) 8.4/8.5 Quiz (5 days) 8.6 Quiz followed by a Unit Test (6 days) Sample Instructional Activities Find the Change Equations of Lines DVD Profits, Variation 1 Proportional relationships, lines, and linear equations 8.EE The Intersection of Two Lines Find rate of change and show that the slope of a line is the same between any two points on the line. Use slopes and intercepts to write and graph linear equations. Develop the slope-intercept form and the point-slope form of an equation. Identify a direct variation. Solve a system of equations by graphing. Standards 8.F.4 8.EE.6 8.EE.5 8.F.3 8.EE.8a Resources Explorations in Core Math Illustrative Mathematics https://www.illustrativemathematics.org/8
iunit: 2.2 Finding Rate of Change and Graphing Linear Equations Using Slope and Intercepts Lesson Focus Lesson 1 of 1 Days: 22 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.F.4 Standard 2: Reason abstractly 8.EE.6 and quantitatively. 8.EE.5 Standard 7: Look for and 8.F.3 make use of structure. 8.EE.8a Standard 4: Model with 8.EE.8c mathematics. 5. Prerequisite Knowledge Ordered pairs Graphing on a coordinate plane Proportional relationships Solving equations 8.1 Graphing Linear Equations 8.2 Slope of a Line 8.3 Using Slopes and Intercepts 8.4 Point-Slope Form 8.5 Direct Variation 8.6 Solving Systems of Linear Equations by Graphing 6. Essential Vocabulary Rate of Change Slope y-intercept x-intercept Slope-intercept form Direct variation 9. Instruction Practices (What are the teachers doing) Lesson on Graphing Linear Equations Show QR Reader video and then read the Essential question. Do Example 1 as a class. Use direct instruction and questioning strategies from the Teacher Manuel as a guide. 7. Possible Misconceptions Using a horizontal change over a vertical change resulting in the inverse for slope. (run/rise) When finding slope using the formula making errors with signs. Suggested Learning Practices How do you find rate of change? How can you show that the slope of a line is the same between any two points on the line? How can you use slopes and intercepts to write and graph linear equations? How can you develop the point-slope form of an equation of a line? How can you identify a direct variation? How can you solve a system of equations by graphing? 8. Teaching Materials Explorations in Core Math Illustrative Mathematics https://www.illustrativemathematics.org/8 10. Learning Practices (What are the students doing) Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how do you find rate of change? Students will be guided by their teacher as they work through Example 1 in their groups. Students will ask and answer questions that arise as they make
Instruct students to do Explore 2. Have students pay special attention to rate of change and unit rate. Have students practice some problems and then close with the Essential question asking students for the key points from the lesson. their way through the process. Students will work on Explore 2 in their groups and discuss questions from the problem. Students will work on their practice problems and then discuss the Essential question; how do you find rate of change? Lesson on Slope of a Line Show QR Reader video and then read the Essential question. Have students do Explore 1. Ask students what the slope represents. (When they complete the problem reinforce that the slope of a line is the ratio of the change in y-values (rise) for a segment of the graph to the corresponding change in x-values (run)). Omit Explore 2 it is very confusing. Have students do Practice problems. Close with the Essential question asking students for the key points from the lesson. Lesson on Using Slopes and Intercepts Show QR Reader video and then read the Essential question. Have students do Explore 1. Before they start let them know the formula they are finding is derived from the formula to find the slope of a line given any two points that lie on the line. (When completed have students highlight the formula with representing the slope and representing the y-intercept). Direct students to do Example 2 and the Try This problems. Direct students to do Explore 3 and pay special attention to the Reflect piece. (it might help some students if they redraw the table vertically) Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you show that the slope of a line is the same between any two points on the line? Students will work on Explore 1 followed by a student-led discussion on some of the questions from the problem. As the students work in their groups discuss the results from the Practice problems. Discuss the Essential question; how can you show that the slope of a line is the same between any two points on the line? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you use slopes and intercepts to write and graph linear equations? Students will work on Explore 1 in their groups. When they complete the task they will highlight the formula Students will work on the Example and the practice problems in their groups and share their work with the class. Students will work on Explore 3 followed by a student-led discussion on the process involved to derive the equation from the table.
Direct students to do Explore 4 as well as the Reflect and the TRY THIS section. (some students may need a hint in finding the points to use from the graph) Direct students to do Explore 5 along with the Reflect part of the problem. Have students do Practice problems. Close with the Essential question asking students for the key points from the lesson. Students will work on Explore 4 followed by a student-led discussion on the reflection piece as well as the problem they tried on their own. Students will work on Explore 5 followed by a student-led discussion on making predictions based on the equation derived. Students will work on practice problems and share their results with the rest of the class. Discuss the Essential question; how can you use slopes and intercepts to write and graph linear equations? Lesson on Point-Slope Form Show QR Reader video and then read the Essential question. Have students do Explore 1. Point out to them to pay special attention to the Reflect piece at the end of the problem. (when through have them highlight the equation for point-slope form Have students do Practice problems. Close with the Essential question asking students for the key points from the lesson. Lesson on Direct Variation Show QR Reader video and then read the Essential question. Have students do Example 1. When completed emphasize the differences in the two graphs. Have students do Practice problems. Close with the Essential question asking students for the key points from the lesson. Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you develop the point-slope form of an equation of a line? Students will work on Explore 1 followed by a student-led discussion on the process involved to derive the equation. They will then highlight the formula for point-slope form. Students will work on practice problems and share their results with the rest of the class. Discuss the Essential question; how can you develop the point-slope form of an equation of a line? Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you identify a direct variation? Students will complete Example 1 in their groups followed by a student-led discussion on the differences in the two graphs. Students will work on practice problems and share their results with the rest of the class. Discuss the Essential question; how can you identify a direct variation? Lesson on Solving Systems of Linear Equations by Graphing Show QR Reader video and then read the Essential Students will watch the video taking notes. The class will discuss the video and the students will highlight the Essential question; how can you solve a
question. Have students do Explore 1. Emphasize checking the solution by plugging it in to both original problems when the graphing is completed. Have students do Example 2. (When students have completed the three graphs reinforce the concept that there may be one solution, infinitely many solutions, or no solutions). Have students compare the differences or similarities in the slopes and y-intercepts of all three graphs. Have students do Example 3. Instruct them to pay special attention to the conjecture in the Reflect piece. Have students do Practice problems. Close with the Essential question asking students for the key points from the lesson. system of equations by graphing? Students will do Explore 1 in their groups sharing their work with the class when they are done. Students will work on Example 2 followed by a discussion on the problems with the emphasis on the slopes and the y-intercepts of the graphs. Students will work on the example with a discussion on the key points of the problem and the conjecture piece. Students will work on practice problems and share their results with the rest of the class. Discuss the Essential question; how can you solve a system of equations by graphing?
Unit: 3.1 Title: Exponents Days: 13 Essential Questions How can you evaluate negative exponents? How can you develop and use the properties of integer exponents? How can you use scientific notation to express very small and very large quantities? How can you add, subtract, multiply, and divide using scientific notation? Content to be Learned Skills 3.1 Integer Exponents 3.2 Properties of Exponents 3.3 Scientific Notation 3.4 Operating with Scientific Notation Assessments 3.1/3.2 Quiz (7 days) 3.3/3.4 Quiz (6 days) Sample Instructional Activities http://www.illustrativemathematics.org/illustrations/395 http://www.illustrativemathematics.org/illustrations/823 Simplify Expressions involving positive and negative exponents. Operations with scientific notation. Standards 8.EE.1 8.EE.3 8.EE.4 Resources Explorations in CORE MATH
Unit 3.1 1. Standards Addressed 8.EE.1 5. Prerequisite Knowledge Order of Operations Meaning of an exponent Base Title: Properties of Integer Exponents 2. Content to be Learned Simplify Expressions involving positive and negative exponents. 6. Essential Vocabulary Integers Exponents Base Power Factors 9. Instruction Practices (What are the teachers doing) Lesson on Integer Exponents Activate prior knowledge and ask students to give some examples of exponents. Could use QR Reader Teacher facilitates as students do Explore 1. Explain that exponents are a shorthand version for writing numbers. Discuss some of the patterns that the students have found. (Teacher may have to cue/hint that as the Lesson 1 of 2 Days 7 Lesson Focus 3. Mathematical Practices Standard 1. Make sense of problems and persevere in solving them. Standard 8. Look for and express regularity in repeated reasoning. 7. Possible Misconceptions Negative Exponents, not being a negative number for ex: is not -8. Any number raised to the zero power is not zero. Some may confuse Product Rule of Exponents with Power Rule of Exponents. Instruction 4. Essential Questions How can you evaluate negative exponents? How can you develop and use the properties of integer exponents? 8. Teaching Materials Explorations in CORE MATH http://www.illustrativemathematics.org/illustrations/395 http://www.illustrativemathematics.org/illustrations/823 10. Learning Practices (What are the students doing) Students will orally give examples of what they know. Students do Explore 1 as a group. Have student-led open discussion on what they may have discovered.
value of the exponent decreases the value of the power is divided by the base. A number with a negative exponent should be written as the inverse of a number with a positive exponent). Once students have found patterns discuss rules ( highlighted in their workbook. Close with Essential Question. How can you evaluate negative exponents? Students will highlight rules in their workbook. Students will answer Essential Question based on what they have learned. Lesson on Properties of Exponents Connect to prior learning by reviewing Commutative and Associative Properties of Multiplication. Explain to them how these properties will make operations with numbers in exponential form easier. Teacher facilitates as students do Explore I. Discuss the three Conjecture pieces from Explore 1 and the importance of the three new Power Rules. Students will write Properties in their notebooks. Students do Explore 1 as a group. Have a student-led open discussion while students highlight three new rules from the Conjecture piece in their workbooks. Remind students as they begin Example 2 of the Order of Operations. Have students work through the example by highlighting the operation they will do first. Close with Essential Question. How can you develop and use properties of integer exponents Students do Example 2A as a group. Students do Example 2B individually. Students will answer Essential question based on what they have learned.
Unit 3.1 Title: Scientific Notation Lesson 2 of 2 Days: 6 1. Standards Addressed 8.EE.3 8.EE.4 5. Prerequisite Knowledge Multiplying and Dividing Exponents Operations in Base 10 Lesson Focus 2. Content to be Learned 3. Mathematical Practices Operations with Standard 2. Model with scientific notation. Mathematics. Standard 5. Use appropriate tools strategically. 6. Essential 7. Possible Vocabulary Misconceptions Scientific Notation Multiplying by means to add 3 zeros? Writing a number in scientific notation and forgetting one of the factors must be; 4. Essential Questions How can you use scientific notation to express very small and very large quantities? How can you add, subtract, multiply, and divide using scientific notation? 8. Teaching Materials 3.3 & 3.4 Explorations in CORE MATH http://www.illustrativemathematics.org/illustrations/476 9. Instruction Practices (What are the teachers doing) Lesson on Scientific Notation QR Reader (optional) video shows a couple examples of scientific notation to standard form. Ask a student to read essential question. Activate prior knowledge of scientific notation before students begin Explore 1. Students may think that a negative exponent means a negative number. Moving decimal points the wrong way!! Instruction 10. Learning Practices (What are the students doing) Students will be watching short video about the lesson while taking notes. Student led discussion on video. Students are brainstorming very large and very small numbers. Students will work in groups to complete the examples in their groups.
Begin Explore 1 with Student led discussion on the Key Concept Box. Example 2 Writing a number in scientific notation. Teacher may want to do an example before starting Ex 2. Have students work through Ex. 2 in groups. Example 3 Students can work in groups to complete Ex. 3a & 3b. Example 4 Comparing Numbers in Scientific Notation Go over the different forms of measurement. Students will work on Explore 1 and then highlight the KEY Concept on Writing Numbers in Scientific Notation. Students will do Example 2. Class will discuss as a group and come up with a rule when translating between scientific notation and standard notation. Students will work on Example 3 and discuss when completed. Students will work on Example 4 and then answer the essential question; how can you use scientific notation to express very large and very small quantities? Lesson on Operating with Scientific Notation QR Reader showing some operations with addition and subtraction using scientific notation. Discuss essential question and then direct students to begin Explore 1. Direct students to try Explore 2. (Remind them about the rule on dividing powers with the same base). Skip Example 3. Direct students to practice 2-3 problems. Close with the Essential question. Students will be watching short video about the lesson while taking notes. Student led discussion on video and what they know about scientific notation. Students will highlight the essential question then work on Explore. Discuss when completed. Students will work on Explore 2 and then discuss their results. Have students show their solutions to their peers. Have students answer the Essential question; how do you add, subtract, multiply, and divide using scientific notation?
Unit: 3.2 Real Numbers, Roots and Pythagorean Theorem Days : 14 Essential Questions How do you estimate and compare irrational numbers? How can you tell whether a number is rational or irrational? How can you use the Pythagorean theorem to solve problems? How can you prove the Pythagorean theorem and its converse? Content to be Learned 3.5 Squares and Square Roots 3.6 Estimating with Square Roots 3.7 Real Numbers 3.8 The Pythagorean Theorem 3.9 Applying the Pythagorean Theorem and its Converse. Assessments Quiz 3-5, 3-6. Quiz 3-7, 3-8, 3-9. Unit 3 Test (Optional) Sample Instructional Activities http://www.illustrativemathematics.org/illustrations/60 http://www.illustrativemathematics.org/illustrations/1417 http://www.illustrativemathematics.org/illustrations/1302 http://www.illustrativemathematics.org/illustrations/1245 http://www.illustrativemathematics.org/illustrations/334 Skills You will find and estimate square and cube roots. Determine whether a number is rational or irrational. Prove the Pythagorean theorem and its converse. Use the Pythagorean theorem to solve problems. Standards 8.EE.2 8.NS.1 8.G.7 8.G.8 8.G.6 Resources
Unit 3.2 Title: Squares and Square Roots Lesson 1 of 2 Days 6 Lesson Focus 1. Standards Addressed 2. Content to be Learned 3. Mathematical Practices 4. Essential Questions 8.EE.2 You will find and estimate square and cube roots. Standard 4. Model with mathematics. Standard 8. Regularity and repeated reasoning. How do you evaluate square roots and cube roots? 5. Prerequisite Knowledge 6. Essential Vocabulary 7. Possible Misconceptions 8. Teaching Materials Operations with Exponents Perfect Squares Number sense, estimation Order of Operations Explorations in CORE MATH Square Root Cube Root Perfect Square Perfect Cube Principle Square Root Irrational Number Rational Number That a square root is half of a number. That a cube root is one-third of a number. That the square roots of certain numbers are between integers. ( is between the integers 6 and 7 and not between 37 and 39) Instruction 9. Instruction Practices (What are the teachers doing) 10. Learning Practices (What are the students doing) Lesson on Squares and Square Roots Activate prior knowledge and ask students what they know about squares and square roots and then read the Essential question. Show QR Reader video. Direct students to begin Explore 1. Pass out cube shaped models ( Discuss the characteristics of their models from worksheet. Direct students to try Explore 2. Direct students to try Example 3. Have students practice some problems and close with the essential question. Students will discuss what they know about squares and square roots and then watch the video taking notes. They will also highlight the Essential question. Students will complete Explore 1 and then discuss as a group. Students will answer questions from the worksheet as they investigate with their manipulative. Students will work on Explore 2 and then discuss as a group. Students will work on Example 3 and then discuss as a group. Discuss results of practice problem and then answer the Essential question; how do you evaluate square roots and cube roots? Lesson on Estimating Square Roots
Show QR Reader video and then read the Essential question. Emphasize the difference between the phrases rational numbers and irrational numbers and give examples of both. (rational numbers terminate or repeat while irrational numbers are non-terminating with the most famous being ) Direct students to begin Explore 1. (May want to do as a class as it is a very difficult concept for most students) Direct students to do Example 2. (Review inequality symbols Direct students to do Example 3 reminding students smaller numbers are to the left of larger numbers. Students will watch the video taking notes. The class will discuss the video and the students will highlight the essential question; how do you estimate and compare irrational numbers? Students will make a 2-column chart which shows examples of rational and irrational numbers. Students will do Explore 1 as a whole class hopefully asking questions as they make their way through the practice. Students will do Example 2 in their individual groups and then discuss their results as a class. Students will do Example 3 in their individual groups and then discuss their results as a class. Have students practice some problems and close with the essential question. Review the results from each of the groups and then answer the Essential question; how do you estimate and compare irrational numbers?